We propose an algorithm for an optimal adaptive selection of points from the design domain of input random variables that are needed for an accurate estimation of failure probability and the determination of the boundary between safe and failure domains. The method is particularly useful when each evaluation of the performance function g(x) is very expensive and the function can be characterized as either highly nonlinear, noisy, or even discrete-state (e.g., binary). In such cases, only a limited number of calls is feasible, and gradients of g(x) cannot be used. The input design domain is progressively segmented by expanding and adaptively refining mesh-like lock-free geometrical structure. The proposed triangulation-based approach effectively combines the features of simulation and approximation methods. The algorithm performs two independent tasks: (i) the estimation of probabilities through an ingenious combination of deterministic cubature rules and the application of the divergence theorem and (ii) the sequential extension of the experimental design with new points. The sequential selection of points from the design domain for future evaluation of g(x) is carried out through a new learning function, which maximizes instantaneous information gain in terms of the probability classification that corresponds to the local region. The extension may be halted at any time, e.g., when sufficiently accurate estimations are obtained. Due to the use of the exact geometric representation in the input domain, the algorithm is most effective for problems of a low dimension, not exceeding eight. The method can handle random vectors with correlated non-Gaussian marginals. The estimation accuracy can be improved by employing a smooth surrogate model. Finally, we define new factors of global sensitivity to failure based on the entire failure surface weighted by the density of the input random vector.

翻译：我们提出了一种算法，用于从输入随机变量的设计域中自适应选择最优点集，以准确估计失效概率并确定安全域与失效域之间的边界。该方法特别适用于性能函数g(x)每次评估代价极高，且函数具有高度非线性、含噪甚至离散状态（如二元）特征的情形。在此类情况下，仅能进行有限次调用，且无法使用g(x)的梯度。输入设计域通过扩展和自适应细化无锁网格状几何结构逐步分割。所提出的基于三角剖分的方法有效结合了模拟方法与近似方法的特性。该算法执行两项独立任务：(i) 通过巧妙结合确定性求积规则与散度定理应用进行概率估计；(ii) 通过新点序贯扩展实验设计。用于未来g(x)评估的设计域点序贯选择通过一种新的学习函数实现，该函数以对应局部区域的概率分类为优化目标，最大化瞬时信息增益。扩展可在任意时刻终止，例如当获得足够精确的估计时。由于在输入域中使用了精确几何表示，该算法对维数不超过八的低维问题最为有效。该方法可处理具有相关非高斯边缘分布的随机向量。通过采用平滑替代模型可提高估计精度。最后，我们基于由输入随机向量密度加权的完整失效面定义了新的全局失效敏感度因子。